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Mirrors > Home > MPE Home > Th. List > Mathboxes > ontgsucval | Structured version Visualization version GIF version |
Description: The topology generated from a successor ordinal number is itself. (Contributed by Chen-Pang He, 11-Oct-2015.) |
Ref | Expression |
---|---|
ontgsucval | ⊢ (𝐴 ∈ On → (topGen‘suc 𝐴) = suc 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suceloni 7517 | . . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On) | |
2 | ontgval 33676 | . . 3 ⊢ (suc 𝐴 ∈ On → (topGen‘suc 𝐴) = suc ∪ suc 𝐴) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ On → (topGen‘suc 𝐴) = suc ∪ suc 𝐴) |
4 | eloni 6194 | . . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
5 | ordunisuc 7536 | . . . 4 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴) |
7 | suceq 6249 | . . 3 ⊢ (∪ suc 𝐴 = 𝐴 → suc ∪ suc 𝐴 = suc 𝐴) | |
8 | 6, 7 | syl 17 | . 2 ⊢ (𝐴 ∈ On → suc ∪ suc 𝐴 = suc 𝐴) |
9 | 3, 8 | eqtrd 2853 | 1 ⊢ (𝐴 ∈ On → (topGen‘suc 𝐴) = suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∪ cuni 4830 Ord word 6183 Oncon0 6184 suc csuc 6186 ‘cfv 6348 topGenctg 16699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-ord 6187 df-on 6188 df-suc 6190 df-iota 6307 df-fun 6350 df-fv 6356 df-topgen 16705 |
This theorem is referenced by: onsuctop 33678 |
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